The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  0  X  2  1  1  X 2X  1  1  X 2X+2  1  1  1  1  1  X  X  X  X  X  0  1  X  2  1  1  X 2X  1  X 2X+2  1  1  1  X  X  X  X 2X+2  0 2X+2 2X  1  X  1  0  X  2  X  X 2X 2X+2  1  1 2X+2 2X+2  1  1  1  1  X  X  1
 0  X  2 3X+2 2X X+2 2X+2 3X  0 3X+2  2 3X 2X X+2 2X+2  X  0 3X+2  2 3X 2X X+2 2X+2  X 3X+2  X 3X  X  0 3X+2 X+2  X  2 3X  X  X 2X X+2 2X+2  X  0 2X  2 3X+2 2X+2  0  X 3X+2 3X  X  2 3X X+2  X 2X  X  X 2X+2 X+2  X  0  2 2X 2X+2  2 2X+2 2X+2 2X+2  0 3X+2 2X  X 3X  X X+2  X  X  X  2 2X+2  0 2X  0 2X  2 2X+2 3X+2 X+2  0

generates a code of length 89 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 88.

Homogenous weight enumerator: w(x)=1x^0+4x^88+94x^89+6x^90+12x^91+3x^92+6x^93+1x^98+1x^110

The gray image is a code over GF(2) with n=712, k=7 and d=352.
This code was found by Heurico 1.16 in 0.5 seconds.